Abstract
Given graphs F, H and G, we say that G is (F, H )v‐Ramsey if every red/blue vertex coloring of G contains a red copy of F or a blue copy of H. Results of Łuczak, Ruciński and Voigt, and Kreuter determine the threshold for the property that the random graph G(n, p) is (F, H )v‐Ramsey. In this paper we consider the sister problem in the setting of randomly perturbed graphs. In particular, we determine how many random edges one needs to add to a dense graph to ensure that with high probability the resulting graph is (F, H )v‐Ramsey for all pairs (F, H) that involve at least one clique.
Highlights
Given graphs F, H and G, we say that G is (F, H)v-Ramsey if every red/blue vertex coloring of G contains a red copy of F or a blue copy of H
For l ∈ N, a sequence of graphs H1, ... , Hl, and a graph G, we say that G is (H1, ... , Hl)v-Ramsey if for every l-coloring of the vertices of G, there is some i ∈ [l] for which G contains a copy of Hi whose vertices are all colored in the ith color
Recall that in our algorithmic proof of the 1-statement in Theorem 1.11, we worked in an ε-regular k-tuple in Gn, using the vertex Ramsey properties of the random graph in each part to iteratively grow a red clique or try to build a copy of H
Summary
For l ∈ N, a sequence of (not necessarily distinct) graphs H1, ... , Hl, and a graph G, we say that G is (H1, ... , Hl)v-Ramsey if for every l-coloring of the vertices of G, there is some i ∈ [l] for which G contains a copy of Hi whose vertices are all colored in the ith color. For l ∈ N, a sequence of (not necessarily distinct) graphs H1, ... Hl, and a graph G, we say that G is Hl)v-Ramsey if for every l-coloring of the vertices of G, there is some i ∈ [l] for which G contains a copy of Hi whose vertices are all colored in the ith color. Hl)-Ramsey if for every l-coloring of the edges of G, there is some i ∈ [l] for which G contains a copy of Hi whose edges are all colored in the ith color. The classical question in Ramsey theory is to establish the smallest n ∈ N such that the complete graph Kn on n vertices is This, the analogous question in the setting of vertex colorings is completely trivial.
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